Regions bounded by a spiral Let Rn be the region bounded

Chapter 8, Problem 41E

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QUESTION:

Let \(R_n\) be the region bounded by the nth turn and the (n+1) st turn of the spiral \(r=e^{-\theta}\) in the first and second quadrants for \(\theta\ \geq\ 0\) (see figure).

a. Find the area \(A_n\) of \(R_n\).

b. Evaluate \(\lim _{n \rightarrow \infty}\ A_{n}\).

c. Evaluate \(\lim _{n \rightarrow \infty}\ A_{n+1}/A_n\).

Questions & Answers

QUESTION:

Let \(R_n\) be the region bounded by the nth turn and the (n+1) st turn of the spiral \(r=e^{-\theta}\) in the first and second quadrants for \(\theta\ \geq\ 0\) (see figure).

a. Find the area \(A_n\) of \(R_n\).

b. Evaluate \(\lim _{n \rightarrow \infty}\ A_{n}\).

c. Evaluate \(\lim _{n \rightarrow \infty}\ A_{n+1}/A_n\).

ANSWER:

Solution 41EStep 1:a) The given logarithmic spiral r = Let be the region bounded by the turn and turn of the spiral in the first and second quadrant for .Let be the area of the region which is bounded by first and second turn of the spiral r = in the first and second quadrant = .Since area of r = f(between r = , r = is given by .Let be the area of the region which is bounded by second and third turn of the spiral r = in the first and second quadrant = .Let be the area of the region which is bounded by third and fourth turn of the spiral r = in the first and second quadrant = .Similarly , ………………Let be the area of the region which is bounded by and turn of the spiral r = in the first and second quadrant = …………….(1) Consider , = , since By , using this result we get , = = - = ( (Therefore , = ( (

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